A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Probabilistic exponential growth model

I have a real valued number $y_t$. At each time step t, $y_t$ is multiplied by $(1 + \epsilon)$ with probability $p$ and multiplied by $(1 - \epsilon)$ with probability $1 - p$. What is the expected ...
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Derivation of first passage time density for Ornstein-Uhlenbeck Process

Consider the following OU process parameterized by two positive real numbers $\alpha$ and $\sigma$: $$dX_t -\alpha X_t dt + \sigma \sqrt{2\alpha} dW_t$$ $$X_0 = x_o > 0$$ We are interested in the ...
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Discrete-time stochastic process - waiting time

Suppose you are given a discrete-time stochastic process which startes in state $C$ and can attain two further states $A$ and $B$. In state $C$ you take two independent exponential times $T_1 \sim E(...
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Kolmogorov backwards equation / stationary distribution

One can in the case of the Fokker-Planck / forward Kolmogorov equation, set the time derivative term to zero, and solve the remaining ODE to obtain the "forward-time" stationary distribution. Does ...
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Show that a permutation $\pi(t,\omega)$ of a ordering relation is $\mathcal{F}_t$

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration, with $\tau < \infty$. The following definition is ...
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How to randomly generate two integer matrices $A$ and $B$, so that entries of 3 metrics $A$, $B$, and $AB$ are within certain range?

I ran into this question when writing a program. I need to generate two matrices, and calculate their product. However, I must ensure all entries are within 8-bit signed integer range, i.e. $[-128, ...
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13 views

Ergodicity in hidden Markov models

Assume that we have a hidden Markov model, where we have a sequence of hidden variables $Z_1, \dots, Z_m$ which form a Markov chain. Now, at each "time point" $i$, an observation $Y_i$ is drawn from ...
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45 views

Suggestions for Constructing a Random Variables from Correlated Observations

Let $\mathcal{X} \neq \phi $ be a finite set. Let $P_{XY_1Y_2}$ be a fixed joint distribution over $\mathcal{X}\times\mathcal{X}\times\mathcal{X}\ $ and that a random sample $(X,Y_1,Y_2 )$ is drawn ...
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Ornstein-Uhlenbeck Process simulation bug

I think I found a bug in a programm somebody posted but I can't fix it. It is about the simulation of an Ornstein-Uhlenbeck Process. The problem is from this [article][1] & and from wikipedia from ...
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Expectation $ \mathbb E^{\mathbb P^{\tau}}_t\left[ X_{\tau}(T)Y_{\tau}\right] $ and measure change.

Under a probability space $(\Omega, (\mathcal F_t), \mathbb P)$ consider two processes $X$ and $Y$ following the SDEs $$ \frac{dX_t}{X_t} = \mu_t \ dt + \sigma(t,T) \ dW^1_t $$ $$ \frac{dY_t}{Y_t} = ...
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Scale invariance of uniform distribution over $\mathbb R^2$?

If we make a uniform distribution of points over $\mathbb R^2$ with 1 point on average per unit square. And we zoom far out and make a density plot (give a color to each cell according to how many ...
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45 views
+50

Understanding the Skorohod-space

I am having a lack of understanding the Skorhodspace considering cadlag processes. A random variable $X$ is measurable mapping between two measure spaces say $(\Omega,\mathcal{F})\mapsto (\tilde{\...
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A proof by René Schilling that a continuous Lévy process is integrable

In his treatise "An Introduction to Lévy and Feller Processes" (arXiv link), Prof. Dr. René Schilling gives a short and seemingly straightforward proof for the claim that a continuous Lévy process is ...
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s.d.e and a bound of $\mathbb E (\lvert X\rvert^2)$ which is not $t$-dependent

Let $X_t=X^x_t$ solution of the s.d.e : $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\ X_0=x$$ Where $b$ and $\sigma$ are 1-lipschitzian. I have proved that : for all $t\geq 0$ it exists $L_t=L_t(x)>0$ s.t. $\...
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Finding g$^∗$, τ$^∗$ in 1-dimensional Brownian motion

How do I find g$^∗$, τ$^∗$ such that g$^∗$(s, x) = sup$_τ$E$^{(s,x)}$[e$^{−ρ(s+τ)}$B$_τ$$^2$] = E$^{(s,x)}$[e$^{−ρ(s+τ^{*})}$B$_{τ^*}^2$ ] , where B$_t$ is 1-dimensional Brownian motion, ρ > 0 is ...
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Supremum of expected value over equivalent measures

I'm not sure how one can proof the following statement: We have a probability space $(\Omega, \mathbb{F}, \mathbb{P})$ and a $\mathbb{F}$-measurable random variable $X$. Furthermore we have a set of ...
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41 views

Show that the only nonnegative superharmonic functions in R are the constants

I am having trouble finding g$^∗$(x) when $$g(x) = \begin{cases} xe^{-x} & \text{for x > 0} \\[2ex] 0 & \text{for x $\leq$ 0}. \end{cases}$$ I would like to use the fact that the only ...
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33 views

Proving that only nonnegative superharmonic functions in R$^2$ are constants

How can I prove that the only nonnegative (B$_t$-) superharmonic functions in R$^2$ are the constants? So far, I know that u is a nonnegative superharmonic function and that there exist x, y ∈ R$^2$ ...
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23 views

Predict the daily usage of Bandwidth of a Network

Context: I want to predict the daily usage of bandwidth of a network (consists a number of users) based on previous use . For example, I want to predict the amount of bandwidth during 8 pm to 9pm ...
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1answer
28 views

Two notions of square-integrability

It seems to me there are two notions for random variables / processes which get labeled square-integrable: $EX^2_t<\infty \; \forall t$ $E \int^t_0 X_s^2 \; ds < \infty \; \forall t$ I ...
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21 views

Expected value of geometric Brownian motion

So everyone knows that the expected value of GBM is given by: $X_0 \exp(\mu t)$ My question is that what does this say about such stochastic processes? Since $X_0$ and $\mu$ are within "my control" ...
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conditional probability on zero probability events and conditional Radon-Nikodym derivatives

Consider a stochastic process $\{x_t\}_{t\in T}$ adapted to some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}\}_{t\in T},\mathbb{P})$ taking values in the state space $(\mathbb{R},\...
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47 views

Expectation of the first passage time of $T_{a,b}$ [duplicate]

Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space and let $W_t$ be standard Wiener process and $$T_{a,b}=\inf\{t:W_t=a+bt\}$$ where $a$ and $b$ are costant.I want to get expectation of $...
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59 views

Correlated brownian motions and Lévy's theorem

$W^{(1)}_t$ and $W^{(2)}_t$ are two independent Brownian motions. How can I use Lévy's Theorem to show that $$W_t:=\rho W^{(1)}_t+\sqrt{(1-\rho^2)} W^{(2)}_t,$$ is also a Brownian motion for a given ...
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15 views

convergence in p-th moment of Euler-Maruyama scheme

I have an SDE with global Lipschitz coefficients on $[0,T]$, i.e. $$ dv = \mu(v)dt + \sigma(v)dW $$ with $|\mu(u)-\mu(v)| \vee |\sigma(u)-\sigma(v)| \leq L |u-v|$ and $W$ being standard 1-d Brownian ...
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19 views

estimation for analytic stochastic processes

this is for experts in probability and stats. There is a theorem, I have seen once: Given a stationary analytic random process, one can show that from the values of a sample path in a finite interval ...
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73 views

Close-form solution for distribution of the stopping time for a path-dependent random process?

A time series $\{x_s\}_{s=1}^{\infty}$ is generated from $N(\bar{x},1/b)\ i.i.d.$. Parameter $\bar{x}$ is drawn from prior distribution $N(\phi_0,1/a)$. Define conditional expectation of $\bar{x}$ as ...
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Ornstein-Ulhenbeck and convergence almost surely

Let this O-U equation : $$ dX_t = \alpha X_t dt + \sigma dB_t,\ \ \ X_0=x $$ where $x,\alpha<0,\sigma$ are constants. I proved that $X_t\xrightarrow{d} X\overset{d}{=}\mathcal N (0, \frac{\sigma^2}...
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1answer
27 views

Probability density function of Poisson Process trajectory

Given a Poisson Process with rate $\lambda$, by a fixed time $t$ we have observed $n$ arrivals at times $t_1 < \cdots < t_n$, with $t_0 = 0 < t_1$ and $t_n < t$ I'm trying to find a ...
4
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1answer
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What Stochastic Calculi Other Than Ito And Stratonovich Exist?

When learning about stochastic calculus, you typically encounter Ito and Stratonovich calculi, usually in that order. There are many differences between the two (Ito processes have better martingale ...
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Stochastic Differential Equation Time-Independent

I know that a generic 1-D SDE can be written in Ito form as: $dX_{t} = \mu(X_t,t)dt + \sigma(X_t,t)dW_t$. I was curious as to how such an SDE is written when modelling time-independent processes. I ...
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1answer
41 views

Expected hitting time of Ornstein-Uhlenbeck process

If I recall correctly, it is known that for a standard brownian motion starting at $0$, that the expected time to hit some level $a>0$ is infinite. I'm curious if there's a proof of what the ...
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Is the integral of an Ito process still an Ito process?

Assume $r(t)$ is an Ito diffusion: $$dr_t = \mu_tdt + \sigma_tdW_t$$ Then, define the following process: $$X_t = \int_0^tr(s)ds$$ Is $X_t$ still an Ito diffusion?
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proof of Markov chain Monte Carlo

This is the first step of proof of MCMC in my notes I have a question, how come $\pi(x)\pi(x_p\mid x)=\pi(x_p)\pi(x\mid x_p)$? Is it true for any markov chains which are ergodic and aperiodic? The ...
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1answer
22 views

$\pi$-system generating cylindrical $\sigma$-algebra

I have stumbled while solving the following problem. It seems simple, therefore your hints would be much valuable. Let $C$ denote the set of all continuos functions $x.$ from $t\in[0,\infty)$ to $\...
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Asymptotic Growth of Markov Chain

I'm interested in the following problem: We have got a time-discrete Markov chain $(X_n)$ with state space $S=\mathbb{R}_+^d$. The transition kernel is discrete in the sense, that for each $s \in S$ ...
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25 views

limit of the absolute value of brownian motion

i'm trying to figure out if the limit of the absolute value of a brownian motion goes to $\infty$ as t goes to $\infty$. from the law of iterated logarithm i know that $\limsup_{t\to\infty} \frac{B(...
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36 views

Conditional Expectation

If we have $Z_{t-1} = (X_1, \ldots,X_{t-1})$ and we know $(S_{t-1} \mid Z_{t-1}) \backsim N (\hat{S}_{t-1}, P_{t-1})$, where $S_t = G_{t}S_{t-1} + w_{t}$ and $X_{t} = F_tS_t +v_t$, so I know that $(...
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Distribution of marginal Wiener process.

Let $(W^1,W^2)$ be a two-dimensional Wiener process with correlation $\rho$. Let $\mathcal{F}^1_t$ be the filtration generated by $W^1$ up to $t$. I would intuitively think that for $h$ measurable ...
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Are there symbolic methods/computing for stochastic processes and stochastic differential equations?

Are there symbolic methods/computing for stochastic processes and stochastic differential equations? Are there some research trends along these lines? Can this be perspective and fruitful endeavour ...
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1answer
59 views

Expectation of Product of Ito Integrals wrt Independent Brownian Motions

Let $W_1(t)$, $W_2(t)$, $W_3(t)$ be independent Brownian motions and $f$, $g$ smooth functions. I want to know if the following is true: $$ \mathbb{E}\left[ \left( \int\limits_0^t f(...
2
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1answer
23 views

Square Integrable Martingales and the Unit Process

Let $X_t$ be a continuous square-integrable martingale. Then it is true (I think, please correct me if I am wrong) that: $$\forall t \in [0,\infty), \quad \int_0^t 1_{[0,t]}(s) dX_s = X_t - X_0$$ So ...
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If a coin toss is observed to come up as heads many times, does that affect the probability of the next toss?

A two-sided coin has just been minted with two different sides (heads and tails). It has never been flipped before. Basic understanding of probability suggests that the probability of flipping heads ...
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1answer
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what's the equilibrium for this special birth-death process?

This is an example from notes I have worked out questions from i to iii, no problems with that. But I don't know how to answer question iv, the note says that "The system reaches an equilibrium for ...
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Question on Brownian motion on unit circle.

I have been trying to investigate some stochastic processes that I find interesting. I came across the Brownian motion on the unit circle. For some reason I would have expected that it keeps "going ...
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+100

Brownian Motion in Confined space, any results?

I am searching for work regarding Brownian motion in a confined space, like a sphere or a cylinder, where the wall will serve as reflection boundary. I am wondering if it is possible to derive results ...
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24 views

When is a stochastic integral a martingale?

In what follows, let the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ as well as the chosen filtration $(\mathcal{F}_t)_{t \ge 0}$ be known, and let $f$ denote an arbitrary locally bounded ...
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2answers
37 views

Why is a continuous Lévy process twice integrable?

In his textbook "Wahrscheinlichkeitstheorie" (de Gruyter, 2008), Jochen Wengenroth shows (p. 144) that if $(X_t)_{t\in[0,\infty)}$ is a continuous, real-valued Lévy process with $X_t\in \mathcal{L}_2$ ...
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How to compute integral of exponential martingale with respect to Poisson process?

Let $N=\{N_t:t\in\mathbb R_+\}$ be a homogeneous Poisson process with intensity $\alpha$ and $M_t=N_t-\alpha t$ the compensated process. I'd like to show that $N$ is not a natural process, i.e. that ...
3
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32 views

Are martingales progressively measurable? (Application to square integrable martingales)

This is an incredibly dumb question, but I'm not sure if I know the correct answer, and it doesn't seem to be stated anywhere on the internet, so here goes: Are martingales progressively ...